2.  A System of equations may be solved exactly by using the substitution method. This method solves one equation so that a variable may be “substituted” in the other equation, which can then be solved for a variable.

3. Y Y = 3x So, anywhere “y” occurs, it can be replaced by 3x

5. This problem does not have something like “y=“ so we must solve for a variable before we can substitute For which variable, from which equation, should we solve? Solve for “x” in the first equation because it has a coefficient of one, and can be used easily for substitution Y=1 Are we done with this work? No What more must we do? Substitute the value of y back into the OTHER equation

7. Which variable did you solve for first, before substituting? Y in the second equation, because it has a coefficient of one Be careful that you solve for positive y, not negative y

8. Y equals more than one number, but the whole expression (3x + 2) can be substituted for y and then solved for x

10. This is a weighted average problem that we solved last semester, here is another way to solve this problem

13. What are the variables in this problem?Buses and minivans

15. When both equations are of the form “y=“ then just set them equal to each other

18. (-4,4) (-9,-7) All Real Numbers Infinitely Many Solutions

19.  How to solve a systems problem, graphically with a graphing calculator  Everyone get a calculator

20.  Clear out old problems from the calculator  Home > 1 > no  Open a graphing calculator screen  2 “Graphs”

21.  Our goal is to find the solution to the following system of equations 2.93x + y = 6.08 8.32x – y = 4.11  First solve each of these equations for y, so the resulting equations can be entered into the calculator for f1(x) and f2(x)

22.  The= solved equations should be as follows y = 6.08 – 2.93x y = – 4.11 + 8.32x  Entered these into the calculator for f1(x) and f2(x)  Cntrl > G if enter line disappears

23.  Your screen should look like this

24.  There is one solution to our problem

25.  The solution to our problem is the point where the two lines intersect. The calculator will help us find that point

26.  Menu > 6 > 4  Use the navpad and the select to select each of the lines, one after the other The intersecting point should be displayed on your screen

27.  The solution to this system is x = 0.906, y = 3.43

28.  To start a new problem on the calculator Home > (right arrow) > Enter (graphs) Complete the problems on the next screen

30.  Solving method for systems of equations when one variable is substituted in for another variable Substitution Method  When the substitution method can be used When a single variable can be solved, that is it has a coefficient of one

31.  Two equations relating to the same variables System of Equations  System of Equations with parallel lines No Solution  System of Equations that are the same line Infinite Solutions  System of Equations that cross at one point That point is the solution to the system

32.  7-2 Substitution Method (2 pages) All Problems